Variable Order Fractional Variational Calculus for Double Integrals

TL;DR

This paper develops a new framework for variable order fractional calculus in two dimensions, including integration by parts, Green's theorem, and Euler-Lagrange conditions, advancing the mathematical tools for fractional variational problems.

Contribution

It introduces three types of variable order partial fractional operators and extends classical calculus theorems to the fractional variable order setting.

Findings

Established integration by parts formula for variable order fractional integrals.

Extended Green's theorem to variable order fractional calculus.

Derived Euler-Lagrange equations for two-dimensional fractional variational problems.

Abstract

We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional Euler-Lagrange necessary optimality condition for variable order two-dimensional fractional variational problems.